2008-03-15
Elena.Wildner@cern.ch
Elena Wildner/ CERN-AT-MCS, Frederick Jones/ TRIUMF (Canada) , Francesco Cerutti/
CERN-AB-ATB
Keywords: Beta Beam, EURISOL, power deposition, heat deposition, FLUKA, ACCSIM
Summary
This note describes studies of energy deposition in superconducting magnets from secondary ions in the “beta beam” decay ring as described in the base-line scenario of the
EURISOL Beta Beam Design Study. The lattice structure proposed in the Design Study has absorber elements inserted between the superconducting magnets to protect the magnet coils. We describe an efficient and small model made to carry out the study. The specially developed options in the beam code “ACCSIM” to track largely off-momentum particles has permitted to extract the necessary information to interface the transport and interaction code “FLUKA” with the aim to calculate the heat deposition in the magnets and the absorbers. The two beta emitters
18
Ne
10+ and
6
He
2+ used for neutrino and anti-neutrino production and their daughter ions have been tracked. The absorber system proposed in the Design Study is efficient to intercept the ions decayed in the arc straight sections as foreseen, however, the continuous decay in the dipoles induce a large power deposition in the magnet mid-plane. This suggests a different magnet design, like an open mid-plane magnet structure (such a magnet has been designed for this purpose) and/or protecting liners inside the magnets. The power deposited in the superconducting magnets is, with the layout proposed in the Design Study, below the recommended value of 10
W/m.
The work described was done in collaboration between CERN and TRIUMF, Canada's national laboratory for particle and nuclear physics, during a 2 month’s visit of one person at
TRIUMF. The work was supported by the European Isotope Separation On-Line Radioactive Ion
Beam Facility, EURISOL, in which “beta beams” is one of the work packages.

1. Introduction
The aim of the beta beam is to produce highly energetic pure electron neutrino and antineutrino beams coming from  -decay of 18 Ne 10+ and 6 He 2+ ion beams, following the reactions
2
6
He
2
 
3
6
Li
3
  e
  
(1) for Helium and
18 Ne 10

10
 18
9
F 9
  e
  
(2)
for Neon. In Figure 1 the Feynman diagram for the beta decay is shown.
Figure 1: Feynman diagram of the beta decay.
Figure 2 : Schematic view of the decay ring with injector chain. The blue arrows from the straight sections of the decay ring point to possible experiments.
The Decay Ring is the final stage of the Beta Beam accelerator complex in the baseline design [1, 2] of the EURISOL Beta Beam Task [3]. 18 Ne 10+ and 6 He 2+ ions would, in this scenario,
- 2 -

be generated in an ISOL 1 front end, accelerated to 100 GeV/u, and ultimately injected into the racetrack structure of the decay ring, where the decays in the long straight sections give rise to well-collimated neutrino beams. The decay products 18 F 9+ and 6 Li 3+ , however, are rapidly lost from the ring and may potentially limit its operation and maintenance.
In the following, we outline the path taken to study these secondary-ion losses via Monte
Carlo methods, starting from beam tracking simulations and proceeding to particle-in-matter simulations.
The decay ring is a race track shaped storage ring, with a circumference of about 6900 m, the same circumference as the CERN SPS. One of the about 2500 m long straight sections, where the useful decay takes place, should in this scenario using 18 Ne 10+ and 6 He 2+ ions, be oriented towards the experimental setup in the Frejus tunnel, some 130 km from CERN. Radioactive ions injected into the decay ring will be a continuous source of decay products, distributed around the ring.
Secondary ions from beta decays will differ in charge state from the primary ions and will follow widely off-momentum orbits (up to 30%, which cannot be easily treated using conventional beam codes, using transfer maps). In the racetrack configuration of the ring, they will be mismatched in the long straight sections and may acquire large amplitudes. 46% of the injected particles are lost during momentum collimation due to the merging process for the injection. The layout of the decay ring will give 16 % of the decay-losses in the two arcs and 38% of the decay-loss happens in the two long straight sections. Only half of the latter can be used for physics: the decay in one of the straight sections pointing to the detector. In this note we study the loss in the arcs. ra ig ig h h t s e e ct ct io io n n
Figure 3: The distribution of the decay in the decay ring.
The decayed ions 18 F 9+ and 6 Li 3+ in the machine have to be managed by introducing dumps and absorbers. These decay products have a different magnetic rigidity from the parent ion beam.
Therefore, the trajectories of the child beams in the magnets are different from those of the parent ions. The child ions are lost along the inside of the machine and may damage the equipment and quench the superconducting magnets. The heat created by the particles is also an extra load on the cryogenic system used to cool the superconducting magnets, which has to be taken into account for the dimensioning of the cryo-system. These problems have been addressed for the design of the decay ring lattice. A first calculation of a large aperture lattice dipole, that is adapted to the
1 Isotope Separator On-Line: an instrument to prepare specific reaction products out of the bulk of products formed in fusion, fission, or multi-nucleon transfer reactions.
- 3 -

conditions in the decay ring, has been made [4]. The quadrupole that has been used in the model is the existing ISR quadrupole [5].
This note describes a first evaluation of the impact of the radioactive decay on the main magnet coils where we use loss maps generated from beam code adapted for simultaneous tracking of particles with very different magnetic rigidity. Previous work for evaluation of energy deposition in the superconducting dipole used a pencil beam of decayed particles impacting the absorber [4]. We base our models and calculations on the optics design described in [2].
We have set up a model of ion decay, secondary ion tracking, and loss detection, which has been implemented in the tracking and simulation code ACCSIM. Methods have been developed to accurately follow ion trajectories at large momentum deviations. Coordinates and momenta of the decayed ions can be detected either at the moment of the decay or at the moment of their impact on vacuum chamber walls so that they can be tracked using other tracking codes, for example particle-in-matter simulations. Using secondary-ion data from ACCSIM, post-processed and interfaced to FLUKA, we have implemented a follow-on simulation in FLUKA with a 3D geometry of decay ring components and physics models for ion interactions in matter, allowing radiological studies and in particular the visualization and analysis of heat deposition in the dipole magnets which is a critical design factor for the ring. In our simulation models i.e. both in
ACCSIM and FLUKA, we have implemented absorber elements [2] which are intended to localize the majority of losses outside of the dipoles. These studies provide estimates of the performance, in terms of loss concentration and management, the effectiveness of absorbers, and the implications for successful superconducting dipole operation.
2. Modelling the decay process
ACCSIM, developed at TRIUMF, is a multi-particle tracking and simulation code for synchrotrons and storage rings. ACCSIM incorporates simulation tools for injection, orbit manipulations, radio-frequency (RF) programs, foil, target and collimator interactions, longitudinal and transverse space charge, loss detection and accounting. The interest for the
EURISOL Beta-beam to use this beam-code is to provide a comprehensive model of decay ring operation including injection (orbit bumps, septum, RF bunch merging), space charge effects and losses (which are 100 % in the beta beam case).
Extensions we had to implement in ACCSIM, needed for the calculations for the Beta-Beam application, are:
• Arbitrary ion species, decay, secondary ions.
• More powerful and flexible aperture definitions (for the absorbers)
• Tracking of secondary ions off-momentum by >30% (unheard of in conventional fast-tracking codes)
• Detection of ion losses: exactly where did the ion hit the wall?
All this is a challenge for tracking with the usual ”element transfer maps”. The new ideas for handling this will be described below.
ACCSIM [6] performs the first stage of modelling the histories of ions, from their injection and stacking in the decay ring, to their decay into secondary ions, and the subsequent and inevitable loss of the secondary ions from the ring, which occurs within 1/2 turn from the decay location. Although ACCSIM can be configured to simulate the entire life cycle of ions: injection,
- 4 -

RF stacking, pre- and post-decay tracking, and loss location, from the tracking and simulation point of view the ions have a dauntingly long lifetime in the decay ring (see Table 1). For the purposes of this study we have therefore considered the steady-state operation of the ring after filling or top-up has occurred, when circulating beam distributions are well-defined. We begin by macroparticle sampling from the phase space of the estimated ion distributions [2]. At the same time an additional coordinate, the ion lifetime, is sampled. Tracking of primary ions is done by element transfer maps, so actual decays are detected via a master clock updated at the end of transport through each element, and then backtracked (by splitting the element transfer map) to the actual decay point inside the element. This allows the precise determination of initial conditions for each secondary ion.
Ion
6
Table 1: Characteristics of ions in the beta beam Decay Ring
He 2+
Charge 
Half-Life in decay-ring Decay Product,
[sec] [revolutions]  p/p
+2 100 80.7 3.50 ∙ 10 6 6 Li 3+ , -0.3338
18 Ne 10+ +10 100 167 7.24 ∙ 10 6 18 F 9+ , 0.1109
3.
Chromatic and geometric effects
The extreme mismatch of the secondary ion rigidity to the decay ring bending strength is beyond the reach of the usual fast-tracking mechanisms (e.g. matrix or thin-lens maps) which are intended to deal with small deviations from the nominal ion trajectory. Relative to the reference orbit of the tracking formalism, the secondary ions are equivalent to particles that are offmomentum by ~10% to ~30% (see Table 1). As a simpler and much faster alternative to direct integration by ray-tracing or high-order map extraction à la COSY [7], we have employed “matrix scaling” techniques to calculate transfer maps for ions of arbitrary large momentum deviation, allowing computation of symplectic dipole and quadrupole maps that are accurate for ions widely off-momentum and off-center from the reference orbit; in essence, ACCIM makes a custom map for each particle.
For quadrupole fields we re-compute the transfer matrix for each particle by scaling the focusing strength
K
1

( K
1
)
0
/( 1
 
) where
( K
1
)
0 is the nominal quadrupole strength and  
 p p is the fractional momentum deviation. This reproduces the (second order) chromaticity of particle tunes and also accounts for optical mismatch (beta beating) in which secondary ions may acquire large amplitudes in the long straight sections and may actually be lost before reaching the arcs.
For dipole fields it is necessary to account for both the different bending radius of the secondary ion, and its entry point to the dipole, which may be far off-center. Both these factors determine the effective length and hence the amount of bending experienced by each ion. For an ion which enters the dipole with coordinates
( x e
, x e

,

) we define a new off-center off momentum reference orbit with entry coordinates as follows: x ref
 x e
, x
 ref

(
  
0
) / 2
- 5 -

Where  is the nominal (on-momentum) bending angle of the dipole, and
0
 is the offmomentum bending angle given by
 
2 sin

1
[(

0
 x ref
) sin(

0
/ 2 ) /

] where 
0

L
0
/

0 is the nominal bending radius and   
0
( 1
 
)
is the off-momentum bending radius. Since the coordinate x '
 dx / ds , the expression for x
 ref is not exact but is well approximated as all angles remain small even for the large along the new reference orbit is L
 
 of the secondary ions. The path length
and the parameters ( L ,

) are used to compute a new transfer-matrix along this reference orbit. The overall transfer map: translation–matrix–translation is thus customized to each particle and has been found to agree well with 8 the dipole in question. th -order COSY maps for
4. Specification of a representative model
The design criteria of the Decay Ring superconducting dipoles [2,4] are strongly codependent with the loss pattern of secondary ions in the arcs. The length of the dipole has been
chosen to be able to efficiently insert absorbers between the magnets, see Figure 4. The loss
pattern can be expressed as a series of Monte Carlo events by ACCSIM, where events arise from sampling of primary ions from phase space distributions, tracking of primary ions through the lattice, sampling of particle life-times and localization of decays, and tracking of secondary ions until they exceed the defined apertures of the ring elements.
Li 3+
F 9+
Figure 4: The dipole length has been chosen to house the absorbers between the dipoles to capture decayed ions in an optimal way. Example for a beam on the central orbit (9 sigma beam size) entering a dipole.
In the present study we have used ACCSIM as an event generator for the code FLUKA [8,
9]. For the beta beams essentially 3 cases of loss can be distinguished. First we have the losses from the RF-merging, taken care of by collimation, secondly extraction and the dumping of the secondary ions produced in the straight sections and, what we are studying here, the particles lost in the arcs.
- 6 -

Figure 5: Loss of ions in the beta beam decay ring lattice (red). The figure is taken from [2]. The chamber size (blue) shows the restricted aperture due to the insertion of the absorbers. The dipoles and the quadrupoles in the arcs are represented in black.
For energy deposition studies we had to find a smallest part of the arc representative of the
repetitive loss pattern. Simulations have shown, see Figure 5, that the loss pattern is repetitive with
the arc cells. Our model can then represent one arc cell. This cell corresponds to a sequence of elements which are tagged as “elements of interest” in ACCSIM. Data from decays in these elements are collected as secondary ion events and constitute the input data to FLUKA (after coordinate transformation, see later).
During ACCSIM tracking, events are recorded for FLUKA input according to two criteria:
1. A secondary ion (from decay upstream) has arrived at the first element in the cell and is within the aperture of the element;
2. A primary ion decays in one of the elements of the cell.
Showers from upstream are here neglected for simplicity, assuming that collimators upstream absorb all (the correctness of this simplification has to be checked in a future more complete model). In both cases, ACCSIM records the event data as follows:
 Turn number, element number, particle number
 Event type (secondary ion or decay of primary)
 Longitudinal position ( coordinate) of the event
 Transverse coordinates
 Ion energy deviation, momenta and reference energy
- 7 -

The particles, 6 Li or 18 F, from these two sets are tracked by FLUKA. At the exit of the cell we score, in FLUKA, the particles coming out of the cell. This should give an indication on upstream showers not absorbed by the absorber system. These make up a third set. The third set is used for crosschecking; we should have roughly the same number of particles entering the lattice cell as particles exiting. If so, our approach is repetitive and the cell can represent the whole machine. The arc-cell is extended by one quadrupole to make the repeatability checks. The particles 6 Li or 18 F, collected at the end of the cell are used as a new input file to FLUKA and should give a similar simulation result as the particles collected in the lattice by ACCIM (the first
set). The model will be discussed in more detail later (chapter 9) and is shown in Figure 14.
5. Overview of the interface between ACCSIM and FLUKA
An overview of the modules, that had to be written to
1. generate the geometrical layout
2. transfer information (coordinates and momenta) between ACCSIM and FLUKA
is given in Figure 6 below.
“Beam Optics”
(Survey data)
Geometry data
FLUKA geo
(input cards)
Geometry generation
ACCIM
(Survey data)
FLUKA geo
(input cards)
ACCSIM
(x,y,s) ACCSIM
FLUKA
Mathematica
(x,y,z) FLUKA
Mathematica
(x,y,z) FLUKA
FLUKA
ACCSIM
(x,y,s) ACCSIM
Particle data conversion
ACCSIM
(x,y,s) ACCSIM
FLUKA
FLUKA
ACCSIM
(x,y,s) ACCSIM
Figure 6: Overview of the routines written to convert data between ACCSIM and FLUKA. Top: the geometry creation, and bottom: the particle data conversions. To the left we see the present situation and to the right the situation where conversions and geometry generation are incorporated in
ACCSIM.
These external modules written for this application in Mathematica (including a beam code
“Beam Optics” [10], also written in Mathematica) should later be included in the ACCSIM code.
For this application we chose to transfer the particle information via files.
6. Model generation
In order to have a simple way of generating the FLUKA geometry of the arc it is necessary to use a survey code due to the fact that ACCSIM uses the conventional curvilinear coordinates following the reference trajectory, whereas FLUKA uses a fixed Cartesian system. Such a survey option is not yet available in ACCSIM. In this application, the survey code of “Beam Optics” gives the coordinates of the reference trajectory and the direction of the particle in a global, fixed,
Cartesian reference system. The survey option gives the (x,y,z)-coordinates and the angle of the reference trajectory, in the global Cartesian system, at the exit of each beam element. For the
- 8 -

dipoles as they are modelled here (front and end faces perpendicular to the magnet axis), one has to remember to correct for the difference between the survey-vector and the vector perpendicular to the end face. The difference is half the bending angle which is to be subtracted from the survey data. This code was used to place and rotate the elements in the FLUKA geometry model.
The extended arc-cell that we used in the simulations can be represented in the following way (most beam-codes express beam-lines as a sequence of predefined elements known to the code):
ArcCell= (S2, QF, S2, B, ABS, B, ABS, QD, S2, B, ABS, B, ABS, QF, S2), where the elements in the cell have the properties as shown in Tables 2 to 4.
Table 2: Data for elements in the optics model of the Beta Beam decay ring arc cell
Element name
S2
QF
QD
B
ABS
Element Type
Straight Section
Quadrupole (F)
Quadrupole (D)
Dipole
Absorber
Length [cm]
200
200
200
568.66
200
Half Aperture [cm]
-
6.2
6.2
8.0+1.0
3.5
QF
QD
Table 3: Quadrupole fields and strengths in the Beta Beam decay ring arc cell
Element name Element Type
Quadrupole (Focusing)
Quadrupole (Defocusing)
Gradient He/Ne [T/m]
45.3285/27.1148
-29.5596/-17.6821
Strength [m -2 ]
0.048483361
-0.031616691
Element name
B
Table 4: Dipole fields and strengths in the Beta Beam decay ring arc cell
Element Type
Bending Magnet
Field He/Ne [T] Length [m] Bending Angle Bending Radius [m]
6.006/3.593 5.6866 
/86 155.669
The FLUKA geometry modelled in SimpleGeo [11] is shown in Figure 6.
y z x
Figure 7: Model of an arc-cell as seen by “SimpleGeo”.
For the dipole, the aperture is taken from [1] with an additional increase of 1.0 cm to compensate for the sagitta, where the sagitta is given by (  is bending radius and  is the bending angle)
- 9 -

s = 
Cos

We suppose, that the dipole is not bent (only 6 m long dipole) and that the dipole, in spite of this, is a sector magnet (as calculated in [1]).
The quadrupole field is given as B z
 *k*z (z being the x or y coordinate and k being the strength), which is used in the FLUKA user-routine “magfld” to define the magnetic field. See tables 2 to 4. Values in the tables have been taken from the Beta Beam official database
( http://beta-beam-parameters.web.cern.ch/beta-beam-parameters/ ).
The beam survey data was cross-checked with another beam-code, DIMAD, with agreement better than 0.01 mm. The lengths of the magnets in FLUKA have been taken simply as the magnetic length. No beam-pipe is modelled: the absorbers fill in most of the longitudinal space between magnets. The absorbers have been shortened by 1 cm at each end, not to have overlap
between the absorbers and the dipoles due to the bending angle, see
Figure 8.
1 cm
1 cm
Figure 8: Absorbers are made 2 cm shorter to have simple modelling and no overlap.
We made two models in FLUKA, the first one by using the explicit geometry specification for all elements in the cell. This approach we checked by using “Simplegeo” [11] for 3D inspection.
A flow chart of the modelling is shown in Figure 9.
Arc cell Lattice
“Beam Optics”
“Mathematica”
Survey data
“Mathematica”
Data for Simplegeo
“SimpleGeo”
Data for Fluka
“Fluka”
Tests +
3D display
Tests + display
Comparison
Figure 9: Flow chart of the steps for the model creation (geometry).
- 10 -

A second model was generated, due to the fact that in a machine containing magnets having the axis not collinear with one of the FLUKA coordinate axes. We would like to make the binning in a cylindrical coordinate system (r,  ,z) with the z axis corresponding to the magnet axis. This is interesting if one wants to conveniently define bins that represent the volume significant for the local heat transport and equilibrium. The approach to this was to use the “Lattice” option in
FLUKA, see the manual on the FLUKA web site 2 . With this option, prototypes of the elements can be defined and placed far from the real model, not to interfere as a physical part of the model.
The “real” element is represented by a “container” which is just a region corresponding to the volume (the geometry of the envelope) of the prototype. By roto-translation of the FLUKA coordinate system, when the particle tracking takes place inside the container, the prototype is placed over the container. In this way the particles are tracked in the prototypes, where details and material are described. For the moment the magnetic field of the magnets has been defined over the container and not over the prototype. This can be improved later by using, in the “magfld” routine (see the FLUKA manual), the common block RTDFCM which provides access to the defined roto-translation variables. In fact using a scoring option USRBIN together with ROTPRIN would have given the wanted scoring. On the other hand the “Lattice” option also permits modification and improvement of the magnet models with a minimum of effort. Both models can be compared for testing and checks by using the total energy deposited in the regions displayed in the standard FLUKA output for the first model and by using special options set up in the “Lattice” model to get this data output. “Beam Optics” produces the survey vector at the end of each element: the coordinates and the angle of the closed orbit, in the global coordinate system. The transformation matrix data used by FLUKA for the roto-translations in the “Lattice” case are automatically generated from the beam code model of the cell.
[cm]
[T]
[cm]
Figure 10: Illustration of the Lattice model of our arc-cell created by FLUKA showing also the Field intensity in T. The prototypes are not part of the machine cell we want to calculate and should be located sufficiently far from the model. The horizontal axis is the z-coordinate the vertical the x- coordinate.
2 http://www.fluka.org/
- 11 -

The roto-translations are defined according to the FLUKA manual where  is the polar angle
(we do not apply it here) and 
is the azimuthal angle (see Figure 11).
y x x ’
 z
Z ’
Figure 11: Definition of the azimuthal angle  for the rotation around the y-axis (a simple rotation around one axis implies zero polar angle)
To define our geometry we rotate, around the y-axis, points in the x-z-plane. The offset is calculated easiest by using the inverse matrix (see the FLUKA Manual under the description of the
ROT-DEFI card). Our transformation matrix for the rotation then becomes
InverseMat rix
Cos

 


0
Sin

0
1
0
Sin

0
Cos


 and the offset needed by the ROT-DEFI option is calculated as
Offset = InverseMatrix * NewCoordinates - OldCoordinates where OldCoordinates are the coordinates at, for example, the theoretical entry of the object itself and NewCoordinates are the coordinates of the prototype, which are placed far from the real model. This offset and the rotation angle from the survey-option are used to create the ROT-DEFI
FLUKA input.
We have to mention here that the Lattice option in FLUKA needs a very good accuracy [10 -8 cm] for the overlapping geometries (prototype and container) to work correctly.
The dipole magnet needs a large aperture and we use the dimensions from the feasibility study in [4]. The quadrupole magnet made for the ISR [5] has been used as a realizable magnet model. The magnets in the first straightforward model and the magnet prototypes in the lattice model are modelled as coaxial cylinders (the vacuum inside then the coil and the yoke). The coil
- 12 -

material has been taken simply as copper for both magnet types, Figure 12. Concerning energy
deposition, we believe this is conservative. We also have the cable insulation and non-filling of the cable (space for Helium cooling). The yoke and the absorber are assumed made of iron.
20.0/20.0
12.0/9.0
Fe
Cu
18.4/12.4
Figure 12: Magnet dimensions in cm, dipole/quadrupole
7. Adapting the reference systems between the tracking codes
We also need the transformations between the reference systems for the coordinate transformations of the particle data from ACCSIM to FLUKA and the inverse (to check the transverse beam ellipse after the FLUKA simulation for example). Accsim uses conventional tracking coordinates ( s , x , x' , y , y' ) where s is the distance along the reference (design) orbit. In a plane perpendicular to the orbit we have the x - and the y -coordinates, where the unit vector of the x -coordinate lies in the horizontal midplane of the ring. The arcsines of the slopes x' and y' given by ACCSIM are the angles between the momentum vector and the survey-vector, the unit vector of the s -coordinate.
The “Beam Optics” code only gives the survey vector at the end of the elements and not at a specific s -coordinate; we adapted the code to give the survey-vector at any s -coordinate. With the survey vector we get directly the x, y and z -coordinates in the FLUKA reference system and we can calculate the direction cosines of the momentum in the following way x f y f z f
 x s
 x a cos(
 s
)

 y s z s
 y a
 x a sin(
 s
)
(3) px f py f pz f
 cos(
 ya
) sin(
 xa

 sin(
 ya
) cos(
 ya
) cos(
 xa
  s
)
  s
)
(4) where x, y , and z with subscript f represent the FLUKA coordinates whereas with subscript s we denote the survey part and with subscript a the ACCSIM coordinates. In our model y s is zero.
 s represents the angle of the survey-vector with respect to the z-axis in the global coordinate system.
The direction cosines of the momentum vector for the input to FLUKA are denoted by p x, py and pz respectively and are the projections of the momentum vector, normalized to a unit vector, on the FLUKA axes.
- 13 -

If a survey option is made available in ACCSIM, the coordinate transformation can be done within that code and will be valuable for its future use in simulations where codes with different kinds of reference-systems are used.
8. Magnetic field routines
The magnet models used in the simulations have been studied to check the feasibility of such magnets, in particular the large aperture dipole [4]. Since the sagitta has been included in the
FLUKA geometry model and the complete field map over the coils and the yoke could not be recalculated due to lack of resources, we have taken a pure dipole/quadrupole field over the apertures only. From earlier studies [12] we have seen that the heat deposition in the coil rather increases very slightly for this approximation of the field, so we consider this approach conservative.
Both models, the model using the FLUKA “lattice” option and the “classical” model, need the magnetic field of the magnets defined over the magnets and not over the prototypes. This introduces, in our case, the complication of hard-coding the roto-translation of the field for both models. However, since in this application we already defined it for the “classic” model, the same field description can be used also for the “lattice” model. The roto-translation data used in the
FLUKA routine for magnetic field calculation was already available from the generation of the
FLUKA geometry model. There are other ways of coding magnetic field routines, as mentioned in chapter 6, by using internal FLUKA variables.
In this application, where very few magnets are implemented and due to strong limitations in
available time for implementations, we used another simplified approach. In Figure 13 we see a
magnet positioned at ( x
1
, z
1
) and rotated by the angle  in the ( x,z )-plane. x y
 z
(x
1
,z
1
) y
0
(x
0
,z
0
) d x local d
(x
2
,z
2
)
Figure 13: A magnet in the global coordinate system of FLUKA, left, and a transverse cut of the quadrupole aperture to the right. To find the field in the roto-translated magnet, we need the FLUKA vertical coordinate y magnet axis.
0
and the distance, d, of the point in the x-z-plane to a line representing the
- 14 -

The vertical dipole field is unchanged for a magnet roto-translated in the horizontal plane, the ( x, z ) plane. The analytical 2-dimensional quadrupole field is defined as
B x
B y

 ky kx
(5) for a normal quadrupole with the axis along the z-axis. If we now roto-translate the magnet in the horizontal plane, the ( x-z -plane) we have to find the distance d , to get the magnetic field from the
2-dimensional field, see Figure 13, right. This distance can be expressed as (refer to Figure 13)
d

(( z
1
 z
2
)( x
2
( z
1

 x
0
) z
2
) 2

( z
2

( x
1
 z
0
)( x
1
 x
2
))
 x
2
) 2
(6) where the entrance and exit-points of the magnet are given in the roto-translation routine
(Mathematica®) in our case.
In the magnetic field routine the simple relations (5) and (6) are used, with the FLUKA y coordinate and with x corresponding to the distance d . Then the field components found in this way are simply projected on the FLUKA axes.
9. Tuning the model
The beam parameters can be seen in Table 5. FLUKA needs the momentum per nuclear mass unit
(nmu). As a reminder, the nuclear mass unit is 1 nmu = 0.931239 GeV/c 2 (1/12 of the 12 C nuclear mass) and the atomic mass unit is 1 amu = 0.931494 GeV/c 2 (1/12 of the 12 C atomic mass). A primary particle in our application is a just decayed ion. We assume the momentum of the parent and the child particles are the same (error less than per mille). In future simulations FLUKA should deal with decay (a decay flag from ACCIM can be put in the file for coordinate transformation).
Beam Decays/s
6 He
6 Li
18 Ne
18 F
[10 9 ]
5.371
-
1.993
-
Table
5: Particle information
Normalization
[GeV/prim.-> W]
Mass
[amu]
Momentum
[GeV/c]
-
0.860
-
0.319
6.019441
6.015125
18.005158
18.001200
560.5257
560.5257
1676.626
1676.626
B


Tm

934.8563
622.7906
559.2624
621.2661
Momentum/amu
[GeV/c]
93.11923
93. 18605
93. 11923
93. 13968
To check the consistency between the beam model and the FLUKA model, for magnetic fields and momenta, tracking was done in both ACCSIM and FLUKA. Tracking of a particle on the reference orbit and of a particle off the closed orbit by 1 mm gave agreement up to 10 micrometers at the end of the arc cell. This error is acceptable over one arc cell.
- 15 -

[m]
Start of cell
[m]
End of cell
Escaping
Decayed in cell
Decayed in machine (with absorbers inserted)
Figure 14: We distinguish 3 sets of particles: particles decaying in the ring (generated and tracked up to the cell entry by ACCSIM for the case all absorbers are inserted), particles decaying in the cell
(generated by ACCSIM) and particles escaping (generated by FLUKA).
10. Scoring particles coming out of the cell
To be able to compare incoming (ACCSIM) and outgoing (FLUKA) particles we have included a special scoring routine based on the fluka routine “fluscw” to score the particles coming into the last quadrupole (the quadrupole belonging to the following arc-cell). The routine is used for the non lattice model and gives particle positions and momenta and the particles can be transferred to ACCSIM for further tracking, if wanted. Our aim was to check that the same amount of particles, within statistical variations, leave the cell as those that are injected. This would give an indication about the repeatability of the approach. We have around 15% of 6 Li particles injected and 12% 6 Li particles coming out of the cell. For the ACCIM simulation, the first cell in the lattice was tagged as the “region of interest”. A better choice would have been to choose a cell a bit farther from the beginning of the arc due to the accumulation of decay in the straight section.
11. Results
The results for the energy deposition in FLUKA are given in GeV/primary. A primary corresponds to the entire ion. The half life of 6 He is 80.7 s and for 18 Ne 167 s at relativistic  =100.
For He we have a decay rate of 8.291∙10 11 particles/s and for Ne 3.079∙10 11 particles/s after a topping up of the machine. If we assume constant energy loss over the machine we will have a fraction cell-length/machine-circumference over one cell. This corresponds to a decay-rate of He in the cell of 5.368∙10 9 particles/s and 1.993 ∙10 9 particles/s for Ne since the cell-length is 44.7465 m and the total machine circumference is 6911.5195 m.
The conversion factors are then
P dep
[ W ]

E dep , FLUKA
[ GeV
 prim

1
]
 rate _ in _ cell [ prim
 s

1
]

1 .
602

10

10
[ J

GeV

1
] (7) which gives for the two species the conversion factors
P dep , He
[ W ]

0 .
860

E dep , FLUKA
and
P dep , Ne
[ W ]

0 .
319

E dep , FLUKA
respectively.
- 16 -

In Figure 15 we see the overall heat deposition pattern projected onto the machine plane
(averaged). The peak heat deposited in the superconducting dipoles can be seen in Figure 17. The
scoring bins in the cable are dimensioned to correspond to thermal equilibrium at steady state.
This volume is taken as the cable cross section times the twist pitch of the cable [13]. The cable is
modelled, for simplicity, as a cylinder. The dipole magnet cross-section can be seen in Figure 16.
We can see that the cable is covering the mid-plane. The energy is deposited in a very small region
around the mid plane, see Figure 18. In Figure 18 we see that the coil is modelled as a cylinder,
compare with the magnet cross section in Figure 16. The value acceptable for a NbTi
superconducting cable for an LHC-like dipole magnet, similar to the one we have used here, is 4.3 mW/cm 3
[14]. We see here that the absorbers are efficient (compare Figure 4). This is consistent
with previous studies [4]. However, with this more detailed study, we see that the particles continuously lost over the second half of the dipole magnet cause peak energy deposition up to
30mW/cm 3 which exceeds the recommended value of 4.3 mW/cm 3 . The relatively high values are explained by the fact that the energy loss is concentrated essentially in the in the magnet midplane. This can be remedied by introducing sufficiently thick liners inside the magnets [15]. We will also study an open mid-plane structure for the decay ring. The power deposited in the
quadrupoles is displayed in Figure 19. The power deposition peaks in the first quadrupole is higher
than for the quadrupoles further in the lattice cell (one order of magnitude), which may be explained by the choice of the arc lattice cell used in the study (the first after the straight section).
This has to be investigated in further studies. For Neon the values are about 3 times higher, which was also indicated in [2].
[cm]
Q1 D1
D2
Q2
A1
D3
A2
D4
A3
Q3
A4
[cm]
Figure 15: Overview picture of the energy deposition in an arc cell (including one extra quadrupole for checking) projected on the horizontal plane for the Li 3+ ions (averaged). Q stands for Quadrupole and
D stands for Dipole. The absorbers (A) are placed after the dipole magnets.
- 17 -

Figure 16: The cross section of the dipole cable structure with the field map for the superconducting magnet used in the study. The mid-plane is covered by the coil. In an open mid-plane magnet there would be a gap in the mid-plane with a wedge (aluminium, stainless steel, or copper) in the place of the coil.
D1 D2
D3
D4
Figure 17: The peak energy deposition in each longitudinal bin along the superconducting cable on the four dipoles in the lattice cell. Recommended limit value for quenching is 4.3 mW/cm 3 . The bars represent statistical errors.
- 18 -

Figure 18: Overview of the energy distribution in the superconducting cable on the dipole 3 in the lattice cell. The superconducting cable is modelled, for simplicity, as a cylinder. Transverse projection, averaged over the length of the magnet.
Q1
Q2 Q3
Figure 19: The peak energy deposition in each longitudinal bin along the superconducting cable on the three quadrupoles in the lattice cell. Recommended limit value for quenching is 4.3 mW/cm 3 . The bars represent statistical errors.
The total heat load of interest for the cryogenic system can be estimated from Figure 20. We
can see that for Helium the total heat load is not exceeding the commonly used reference value 10
W/m [13].
- 19 -

Figure 20: The total power deposition in the absorber, yoke and coil regions of the superconducting magnets (Li 3+ ions). Recommended value for the cryogenic system is 10 W/m. “Particles re-injected” are particles coming out of the cell in the beam pipe and re-used in a new simulation as particles coming into next identical cell. These particles should be absorbed at the first absorber (this is the case) but the energy deposited in the first absorber is larger than for the particles coming from the machine. These issues should be further studied (simulate particle decay in cells not at the immediate entrance of the arc)
12. Future developments
We see from this short study that the energy deposited in the super-conducting magnets seems to be manageable with some modifications of the magnet shielding. The interface between
ACCSIM and FLUKA has to be re-evaluated and integration of the codes has interest for future studies of decay in the beta beam decay ring. Using the open mid-plane superconducting magnet designed for this purpose [16] and/or thick absorbing liners inside the magnets instead of the absorbers between the magnets seems to be reasonable options for future studies. Future studies on energy deposition in the decay ring magnets will be based on a new adapted beta beam scenario with other ion species having hopefully better production rates (EUROnu framework, FP7).
13. Acknowledgements
We acknowledge the financial support of the European Community under the FP6 “Research
Infrastructure Action – Structuring the European Research Area” EURISOL DS, Project Contract
No. 515768 RIDS.
Pierre Delahaye, CERN, calculated the masses (in nmu) displayed in table 5.
- 20 -

14. References
[1]
First design for the optics of the decay ring of the beta-beams, A. Chancé and J. Payet,
EURISOL DS/TASK12/TN-06-05
[2]
Simulation of the losses by decay in the decay ring for the beta-beams, A. Chancé and J.
Payet, EURISOL DS/TASK12/TN-06-06
[3]
The Beta-beam within EURISOL DS, M. Benedikt and M. Lindroos (doc-version), EURISOL
DS/TASK12/TN-06-01
[4]
E.Wildner, C. Vollinger, “A large aperture superconducting Dipole for Beta Beams to minimize heat deposition in the coil”, Particle Accelerator Conference 2007, Albuquerque, New
Mexico, 2007
[5] J. Billan, R. Perin, L. Resegotti, T. Tortschanoff, R. Wolf, “Construction of a prototype superconducting quadrupole magnet for a high luminosity insertion at the CERN Intersecting
Storage Rings”, CERN 76-167
[6]
F. Jones, “Development of the ACCSIM tracking and simulation code” , Particle Accelerator
Conference 1997, Vancouver, Canada, 1997
[7]
M. Berz, Computational aspects of design and simulation: COSY INFINITY, Nucl. Instrum.
Meth. A 298:473, 1990
[8] FLUKA: a multi-particle transport code:.A. Fasso`, A. Ferrari, J. Ranft, and P.R. Sala,
CERN-2005-10 (2005), INFN/TC_05/11, SLAC-R-773
[9] The physics models of FLUKA: status and recent development,
A. Fasso`, A. Ferrari, S. Roesler, P.R. Sala, G. Battistoni, F. Cerutti, E. Gadioli, M.V. Garzelli, F.
Ballarini, A. Ottolenghi, A. Empl and J. Ranft, Computing in High Energy and Nuclear Physics
2003 Conference (CHEP2003), La Jolla, CA, USA, March 24-28, 2003, (paper MOMT005), eConf C0303241 (2003), arXiv:hep-ph/0306267
[10]
Autin, Bruno (ed.) ; Carli, Christian (ed.), D'Amico, Tommaso Eric (ed.), Gröbner, Oswald
(ed.), Martini, Michel (ed.), Wildner, Elena (ed.) et al., “Beam Optics : a program for analytical beam optics”, CERN-98-06. - Geneva : CERN, 1998
[11]
Theis C., et al., "Interactive three dimensional visualization and creation of geometries for
Monte Carlo calculations”, Nuclear Instruments and Methods in Physics Research A 562, pp. 827-
829, 2006
[12] Christine Hoa, Francesco Cerutti, Elena Wildner , “Energy Deposition in the LHC Insertion regions IR1 and IR5”, LHC Project Report, to be published
[13] P. Limon, Fermilab, private communication
[14] LHC Design Report, Vol. 1, page 157, ISBN 92-9083-224-0
[15] E. Wildner, C. Hoa, E. Laface, G. Sterbini, “Are large aperture NiTi magnets compatible with 1 e35 ???”, CERN-Yellow Report, to be published.
[16] J.Bruer, private communication
- 21 -